What would Sophia Loren make of Aman Sveti Stefan, I wonder? The Italian superstar was a fan of the 1950s hotel that was once on this Montenegrin rock. It had a fleet of Chevrolet cars to transport guests across the causeway and around the coastline, and served elaborately embellished lobster.

That's all gone, replaced by the austere lines that define the Aman resort aesthetic (present across the globe from Cambodia to the Caribbean). Bare stone walls in bedrooms, rough ceramic shampoo bottles and rustic village salad on the menu signal a major change. But if the glitz has been replaced by restrained tones, make no mistake – it's still for the wealthy, it's just that now it's stealthy.

The rocky outcrop that is Sveti Stefan is Montenegro's most famous view. It may actually be its only famous view. If you can place the country on a map you're already in a well-informed minority. The country gained independence from Serbia in 2006 and is trying to build up a tourism trade. The easiest access is from Dubrovnik, where today Wizz Air starts flying from Luton.

So you make it into this small, vertiginously mountainous country and arrive at its one internationally known site. What can you expect? I had no idea, being an "Aman virgin". There are such people as "Aman junkies" – wealthy travellers who always make a point of visiting each new resort that opens its discreet, unsigned doors. They have been waiting for four years for Aman Sveti Stefan. I arrive on day seven of the village's new incarnation.

It's a remarkable approach. Walking across the causeway, gazing up at the rocks and rustic stone walls that ring the island, it looks as if it's guarding a secret at its centre, its back turned to the coast (and it's true that everyone who has inhabited the island is a clique, from the 12 families of the Pastrovici tribe back in the 1400s to the glossy posse of the 1960s and 1970s, to today's touring elite). One of Sveti Stefan's features is that once inside, its higgledy-piggledy lanes lead you up and downhill, round and around, and there is no centre, just more scattered stone cottages and the odd church. Oh, and a dramatic black-tiled clifftop pool too. There is a piazza, where guests can eat pizzas, antipasti and other posh, rustic food at artfully strewn tables and chairs – but that is stumbled upon, rather than signposted.

In fact, nothing is signposted. I'd know that if I wasn't a first-timer. Henry Gray, Aman Sveti Stefan's ebullient general manager, fills me in over dinner. He's worked with Aman for many years, and discretion to the point of obliqueness is a company trademark. In fact, the "street" signs on Sveti Stefan are a huge departure from the norm (not that they help terribly much, there's no map and it takes day or two before you can confidently find your way from reception to your room).

He and his wife/co-manager Char epitomise the open-shirt, flat-sandalled chic that is Aman. The shirt will, of course, be crisp and tailored, the sandals designer-labelled.

The rooms: well, no two are the same. Because Sveti Stefan is a protected site, the buildings are largely the same ones that once were home to the Pastrovici tribe, and Aman had to work within draconian building restrictions. Interior designer Marylou Thomson – who has been with the project since Aman first signed the 40-year-lease in 2007 – explains. "The honesty of this place means that if you do something wrong, it sticks out like a sore thumb." She laughs, but installing the super-high-spec features demanded by Aman guests (and expected at the super-high prices) cannot have been easy. The rough plaster walls and simple timber or stone floors in each room are immaculate, and under that shower, the stone floor is heated.

Marylou had to replace one in three of the original beams that are a feature of many of the rooms, the legacy of a 1979 earthquake which destroyed many of the buildings along Montenegro's coastline. I'm not confident that I'd spot the difference, such is the attention to detail. Furniture is simple, Shaker style, made in Serbia from Slovakian oak. The lanterns that hang in each window – a nod to the fishermen's wives who did the same to light their husbands home from the sea – may look simple, but they're a world away from your standard storm lantern.

"We wanted it to look like a village, how it was intended to be," says Thomson. Her jaunty tone belies the 100 straight days of rain that blighted the main building work in 2009. We're sitting under a loggia at The Olive Tree, one of the planned 11 restaurants at the resort (some areas will not be completed this season, and will open next spring). It's on the mainland, on the first of three beaches that face the island.

The Aman resort encompasses three bays and the 1930s Villa Milocer building, with its eight suites and stunning wisteria-clad seafront loggia. Breakfast Montenegrin-style is not for the faint-hearted, but we give it a go. There's prosciutto, herb-studded njeguski cheese, cheese pie, dinky cornbread muffins, a sauce that is raw whey, tiny doughnuts and twisted sweet pastries, which our waiter describes as "fast bread. Eat with the domestic apricot." He brings us the traditional glass of rakija (local grappa). Nine o'clock seems a little early, so we stick to the freshly squeezed juice.

As we talk about the project – how it has inched forward over the four years of her involvement, with debates such as "tablecloths or no tablecloths?" taking some time – people start to pass by, creating a steady stream of beach-bound families and couples. They can't all be staying at the Aman?

Of course they're not. Part of the agreement between the government and Aman is that the public are allowed access to the beaches, and to walk the hilltop paths that link them (the village is just one part of a 32-hectare Aman property). However, they will have to pay a fee to use the beaches, and are not allowed on to Sveti Stefan itself unless they book a restaurant table.

"There is some local resentment," acknowledges Marylou. "They used to be able to tour the island for €5; now they're not allowed. But those who have seen the sensitive way the site has been treated have relaxed... It's still prickly around the churches."

Yes, the churches. Sveti Stefan has three, two dedicated to the saint who gives the island its name, and one Church of the Transfiguration. They are locked and the keyholder is the local priest. If his congregation are not allowed across the isthmus, then hotel guests aren't allowed to poke around the church's treasures. But the relationship seems to be a cordial froideur, if such a thing exists. The legions of builders who restored the stone cottages did much to strengthen and tidy up the churches' exteriors too. And the doors are flung open on high days and holidays.

Standing at the door to the oldest one, from 1464, with its staggeringly beautiful view over the bay and its soundtrack of waves crashing on the rocks below, I'd love to have got married here. I can imagine very wealthy families occupying the entire 50 rooms and creating a wedding wonderland, a thought that has clearly occurred to Aman.

It's difficult to stir myself from the vast, insanely comfortable bed in my room to do much exploring. The thick white walls, putty-coloured curtains and leather-edged sisal rugs create a sensory comfort blanket – and the subtle Bose hi-fi, giant bathtub and handmade linen slippers certainly help. If I were so inclined, I'd want to put everything – from the white linen robe, to the navy rope bag for guests to transport their iPods and books to the pool or beach, to the delicate Balinese porcelain pots for toiletries – into my suitcase. There's no shop to purchase these luxury goodies at this stage, but I'm told one is planned. Wise move.

Somewhat reluctantly, I stroll over the causeway and into a car with Marco, one of the hotel's guides. He's bursting with pride about Montenegro. We tour Kotor, a Unesco-listed medieval walled town of narrow, cobbled pathways and the occasional square with a church or museum to gaze upon. Its walls extend up the steep mountain behind. Hardy souls can walk up to the crumbling remains of St Ivan's Castle at 260m and look down at the luxury yachts and expanse of fjord on which Kotor sits.

I get into a little boat and travel across the silent, crystal clear water to the twin tiny islands that sit between the impossibly pretty town of Perast (where the influence of the region's Venetian rule between around 1400 to 1800 is very clear) and the narrow neck that separates the bay from the rest of the fjord and the Adriatic sea beyond.

"Our Lady of the Rock" is a man-made island, in the shape of a ship. It was created over 200 years starting in 1452, built up from a natural rock of just a couple of square metres by fishermen who wanted the protection of the Virgin Mary. They sank vessels here, more than 100 of them, and cast rocks and stones into the bay until they reached the surface, then created a church. I study the silver votives left by grateful sailors, the dried flowers and lace of long-ago brides, left by tradition in thanks – and wonder at the quiet, careful pride Montenegrins have for their natural and spiritual treasures.

Lunch at a local waterside restaurant Stari Mlini is bliss. As you'd expect, fish is what's good everywhere. Here, cuttlefish risotto (near neighbour Italy influences much of the cuisine) and octopus baked in ash are two high points.

Budva, back down the coast towards Sveti Stefan, is also an ancient walled town, but although the old settlement shares the church spires and worn-smooth cobbles of Kotor – and nuns selling the same fabulous soap that appears in the Aman bathrooms – outside the walls, it's a thoroughly modern resort. Russian-built black-glass hotels sprawl, superyachts bob cheek-by-jowl in the harbour and plenty of bars, restaurants and shops entertain those that descend on this tourism hub in high season.

I buy a few postcards and head back to the calm of Sveti Stefan. Dinner is at the Queen's Chair, the restaurant perched high above the Queen's beach: a tiny, immaculate horseshoe of sand with fig and olive trees at its fringes. Here the riches of Montenegro's coast are given haute-cuisine treatment: sea bass, sea bream and scallops are all spankingly fresh and carefully cooked.

Perhaps the cleverest asset this Aman has is its staff. Not just the management – who live and breathe the stratospheric standards of the company – but the 20 or so men and women who worked in the old hotel, who remember when Yugoslavia's President Tito came to stay at the Villa, and when Princess Margaret and Lord Snod (their take on Snowdon, which I don't correct, finding it too sweet) crossed the isthmus.

If they served Sophia Loren and Carlo Ponti in their suite – which today is the Sveti Stefan suite, complete with its own pool and clifftop terrace, yours for €2,500 a night – they're not saying. Discretion reaches back beyond Aman days, and the care and attention they show guests – from quietly placing my handbag on a chair to explaining the components of the dainty, delicious antipasti – suggests a lifetime of training.

Another excursion that matches the blissed-out atmosphere on Sveti Stefan, meanwhile, sees me kayaking on Lake Skadar, the largest body of water in the Balkans. This vast, remote lake is a national park, protected so that the myriad birdlife can live in peace. I slip into the water some 17km away from the main lake, at the village of Rjeka Crnojevica – where, fact fans, the first coins in Europe were minted – and spend several hours with guides Amy and Paul, who point out the animals, flora and fauna so abundant around me.

On my last night on Sveti Stefan, I quiz the deputy minister for tourism, Nebojsa Popovic, on Montenegro's future. He talks enthusiastically about the arrival of other luxury hotel chains. He has a brilliantly straightforward approach. "We have lots of military property that we don't need, so the government took the good decision to make them into tourism locations... The military always have the best locations!"

I've seen just a tiny section of Montenegro. There are fledgling ski resorts and 6,000km of new hiking and biking trails. There are mountain towns and ancient monasteries to visit. There's the region's spectacular wines to sample – and if it's all like the utterly delicious Status Barrique, Milovic 2007, there are treats in store. I intend to come back before long, before the budget airlines increase the 600,000 population of the country each summer to creaking point.

Whatever happens, today's celebrities – in search of ultra-privacy and discreet luxury – will surely be beating a path to Sveti Stefan. Just as long as they leave their Louboutins at the door.

Travel essentials: Montenegro

Getting there

* The capital, Podgorica is served by Montenegro Airlines (020-7864 4031; montenegroairlines.com) from Gatwick. Alternatively, Sveti Stefan can be accessed from Dubrovnik in neighbouring Croatia. Wizz Air (0906 959 0002; wizzair.com) launches a new flight to Dubrovnik from Luton today. The city is also served by easyJet (0905 821 0905; easyJet.com) from Gatwick and Stansted; British Airways (0844 493 0787; ba.com) from Gatwick; Jet2 from Leeds/Bradford, Belfast, Manchester and Edinburgh; FlyBe (0871 700 2000; flybe.com) from Birmingham; and Bmibaby (0871 224 0224; bmibaby.com) from Birmingham.

* Original Travel (020-7978 7333; originaltravel.co.uk) offers three-night trips to Montenegro from £1,602 per person. The price includes return flights from Gatwick to Podgorica, private transfers, three nights' accommodation at Aman Sveti Stefan, a private tour of Kotor Old Town, and a kayak excursion on Lake Skadar.